A type of mathematical model that has been
developed for sumerged round buoyant jets is the length-scale model.
Discharges flows can be divided into different regimes each dominated by
particular flow properties. Within each regime, the flow may be approximated
with simple mathematical relations describing the simplified problem. A
model that uses asymptotic solutions is refered to as length-scale model
because of length scales to delineate the extent of the regimes for which
the mathematical expressions are valid. The pollutant concentration, in a
certain instant, and at a distance x (meters) in the X-Axis and at a
distance y(meters) in the Y-Axis will be given by:

c =cc exp[-(r/b)2] (1)

where c is the pollutant concentration, r is the
distance from the point (that we are calculating) to the center of the line
that forms the polluting plume, cc is the pollutant concentration
in the center of the plume line and b is the plume half-width. We attempt to
link the momentum dominated and buoyanvy dominated regimes into one
relationship by using proposed relations for the transition where:

z/Lb =24/3[(1/2)(x/Lb)2+(Lm/Lb)(x/Lb)]1/3
(2)

b/Lb =cb[(1/2)(x/Lb)2+(Lm/Lb)(x/Lb)]1/3
(3)

S=cs(uo/ua)[(1/2)(Lb/Lm)(x/Lm)2+(x/Lm)]1/3
(4)

Lb=plume-to-crossflow length scale

Lm=jet-to-crossflow length scale

x=horizontal downstream coordinate in global
coordinate system

y=horizontal coordinate in coordinate system
perpendicular to ambient crossflow

z=vertical coordinate

ua=ambient velocity

uo=discharge velocity

S=dilution along the plume centerline C/C0
being C the centerline pollutant concentration and the C0
initial pollutant concentration at the discharge.

cb=constant of proportionality that
can be modified by the user (can be determined experimentally)

cs=constant of proportionality that
can be modified by the user (can be determined experimentally)

cxy=constant of proportionality that
can be modified by the user (can be determined experimentally)

We obtain solutions
for a vertical buoyant jet in a crossflow.
And buoyant
jets discharged horizontally perpendiculat to crossflow.

z/Lb =cxy(x/Lm)1/3
(5)

This model performs satisfactorily for simple
flows with no shoreline interaction or attachment. Strong crosscurrents or
limited depths causing attachment with the downstrean bank or strong initial
buoyancy render this model invalid. In addition, they are incapable of
simulating any far-field processes that occur after a certain distance.

The stratification phenomena is the existence of two homogeneous water
layers and separated by a thin thermocline layer. In such a case, we can say
that the water is stratified. There is no exchange of pollutants through
this picnocline layer.
In case we suppose that a picnocline layer exists, we will be able to check
the stability by means of the application of the following equation:

[u02 B+ Ua2 H]/[(u0 B
g’)2/3 H] <0.54

u0=effluent velocity (m/s)

H=water depth at discharge position (m)

g’=reduced gravity acceleration (m/s2), g’=g(ρa-
ρ0)/
ρ0

g=9,81m/s^2 (gravity acceleration)

ρa=water
density (Kg/m3)

ρ0=effluent
density (Kg/m3)

Typical values (pollutants):

Organic matter as DBO5 - 350g/m3

Suspended matter - 600g/m3

E. Coli - 1012 /m3

N2 (total) - 30 gN/m3

Effluent velocity - entre 0.6 y 0.8 m/s

Port diameter - 6cm

Dispersion coefficients:

Horizontal dispersion:
Ky(m2/s)=3x10-5 B4/3.

B=initial plume width(m)

Vertical dispersion:
Kz(m2/s)=4x10-3 Ua e

e=
thickness of the mixing layer

Ua=horizontal ambient velocity (m/s)

2.1 Water is stratified

2.1.1 Multiport diffuser.-
We have three different cases:

Case I:

θ
>=65º F<=0.1 ó

θ
<65º F<=0.36

θ=angle
between Ua vector and diffuser.

F=Fraude number F=Ua3 (g’q)-1

q=Unitary flor rate in the diffuser q=QLt-1 (m2/s)

Q=Discharge flow rate (m3/s)

Lt=Diffuser length (m)

S=initial dilution

In such a case, we have the next relationships

S=0.27 Ua H q-1 F-1/3

e=0.29H

B=SQ/eUa

Case II:

25o=<θ
<65º F>0,36 (*)

In such a case, we have the next relationships

S=0.38 Ua H q-1

B=max[Ltsin
θ;
0.93Lt
F-1/3]

e=SQ/BUa

Case
III:

θ
<25º 0,36<F=<20

In such a case, we have the next relationships

S=0.294 Ua H q-1 F-1/4

B=max[Ltsin
θ;
0.93Lt F-1/3]

e=SQ/BUa

Case
IV:

θ
<25º F>20

In such a case, we have the next relationships

S=0.139 Ua H q-1

B=max[Ltsin
θ;
0.93Lt
F-1/3]

e=SQ/BUa

Case
V:

θ
>65º F>0,1

In such a case, we have the next relationships

S=0.58 Ua H q-1

B=max[Ltsin
θ;
0,93Lt F-1/3]

e=SQ/BUa

From Case II to Case V, and if e>H, we take e=H and S= UaBH/Q.

2.1.2 Separated ports.-
We will solve this case by means of a iterative mathematical method

B=max[Ltsin
θ;
0,93Lt F-1/3]

S=0.089 g’1/3 (H-e)5/3 Qb-2/3
(***)

e=SQ/BUa

Qb= flow rate at each single port(m3/s).

The number of iterations can modify by means of the parameter N_it of the
function Calculation parameters of the program.

Increasing N_it value, we increase the numeric convergence but we will need
more time of calculation. We should look for an optimized value of N_it.

Single port.-
In such a case, we have the next relationships

e=0.15H

S=0.089 g’1/3 (H-e)5/3 Q-2/3

B=SQ/eUa (*)

However, at high velocity values and if B<=0.3H, the approximation is not
correct .

2.2 Water is not stratified

In this case, the picnocline or thermocline has been formed. We will
distinguish the following cases:

2.2.1 Multiport diffuser.-
In such a case, we have the next equations

ymax=2,84 (g’q)1/3 Г -1/2

S=0,31 g’1/3 ymax q-2/3

B=max[Ltsin
θ;
0,93Lt F-1/3]

e=SQ/BUa

where Г=-(g/ρ)dρa/dy is the stratification coefficient (s-2)
and ymax is the thickness of the mixing layer (m).

2.2.2 Separated ports.-
In such a case, we have the next equations

ymax=3,98 (g’Qb)1/4 Г -3/8
(***)

S=0,071 g’1/3 y5/3max Qb-2/3

B=max[Ltsin
θ;
0,93Lt F-1/3]

e=SQ/BUa

2.2.3 Single port.-

ymax=3,98 (g’Q)1/4 Г -3/8

S=0,071 g’1/3 y5/3max Q-2/3

e=0,13 ymax

B=SQ/eUa

For a profile of velocities different from the previous ones, it will be
required a more complex method of numeric integration to solve the problem.

2.3 Near mixing zone and distant mixing zone

We need to know the place where the plume centerline crosses the water
surface or picnocline layer. To calculate this point we will use Ua
and the vertical velocity

Multiport diffuser.-
W=1,66(g’q)1/3 being W the vertical velocity of the effluent
(m/s).

Separated ports.-
W=6,3(g’Qb/H)1/3 .

Single port.-
W=6,3(g’Q/H)1/3.

In the last two cases, H will be replaced by ymax when the water is
stratified. The point localization with regard to the place where the plume
centerline crosses the surface, gives us the near and distant mixing zone
definitions.

2.4 Concentration calculation

The concentration value in a plume point is determined by the X,Y,Z
coordinates and is given by the equation:

C(X,Y,Z)=(C0/S) F0(t)F1(t)F2(Y,t)F3(Z,t)

C0=pollutant concentration in the effluent

S=initial dilution

being t=X/Ua. F0(t) takes into account
non-conservative pollutants and is equal to:

F0(t)=10-t/T90

The F0, F1, F2 and F3 functions
depend on being in near or distant mixing zone.

(a)Near
mixing zone:

In such a case, the equations are

F1(t)=1

F2(Y,t)=(1/2)[erf[(B/2+Y)/(σy21/2)]+
erf[(B/2-Y)/(σy21/2)]]

F3(Z,t)=(1/2)[erf[(e+Z)/(σz21/2)]+
erf[(e-Z)/(σz21/2)]]

being
σy=(2Kyt)1/2
and
σz=(2Kzt)1/2.
The program calculats erf function by numerical integration. The the
precision of integration method depends on the parameter N_int. Increasing
N_int value, we increase the numeric convergence but we will need more time
of calculation. We should look for an optimized value of N_int.

(b)Distant
mixing zone:

In such a case, we approach

F1(t)=(2π)-1/2B σy-1/2

F2(Y,t)=exp[(-Y2/2σy2)]

F3(Z,t)=e/Hh

being
σy=(B2/16+2Kyt)1/2.
Here, we suppose that the plume was homogenized vertically when the water
depth was Hh, that is the depth in the point where the thickness
of the plume begins to occupy the whole layer of water. The program
calculates considering the bottom of the sea like a flat surface. Then, Hh
is the water depth at the location of the deepest outfall. If you want to
consider a higher water thickness than outfall depth, you can draw a deeper
outfall whose pollutant concentration is null. In asuch a case, the water
depth is the depth of the deepest outfall. The calculation will not be
affected by the null concentration of the deepest outfall.

Errors
and comments in the model:

(*) We have found, in our opinion, typographic
errors in
Orden del 13 de Julio de 1993 del Ministerio de Obras Públicas y Transportes
del Reino de ESPAÑA, B.O.E. Martes 27 de Julio de 1993, página 22861
that we have corrected considering mathematical consistency. The software
assumes the present corrections in the calculation.

(**) Important
note for DESCAR 3.0 (or lower versions):

In the
approved model, T90 is un hours (this is used by the program). However,
and in equation F0(t)=10-t/T90
of the approved model(1), time must be in seconds . Following criteria of
mathematical coherence and results
T90 must be expressed in seconds (multiply by 3600 seconds in one hour).
At this point, the user can work following two different ways: using the
approved model as is or rectify in the T90 input data. For example, for
a T90=2
hours value, the user can introduce as input data:

(***) In the
model, it is not found a relationship between
Qb y
Q. This relation must be n
(number of ports). The calculations assumes that
Qb y
Q are the same (that is always true for a single port). You can introduce a
number of ports in the model parameters window.

IKSR 2000 : M. Braun, “The Pathways for the most important hazardous
substances in the rhine basin (during floods)”, International Commission
for the Protection of the Rhine, Koblenz, Germany, in Int. Symposium on
River Flood Defence, Kassel, Kassel Reports of Hydraulic Engineering No.
9/2000